Properties

Label 471.2.q.b
Level $471$
Weight $2$
Character orbit 471.q
Analytic conductor $3.761$
Analytic rank $0$
Dimension $336$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 471 = 3 \cdot 157 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 471.q (of order \(39\), degree \(24\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(3.76095393520\)
Analytic rank: \(0\)
Dimension: \(336\)
Relative dimension: \(14\) over \(\Q(\zeta_{39})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{39}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 336 q - 2 q^{2} + 14 q^{3} - 26 q^{4} + 2 q^{5} + q^{6} - 4 q^{7} + 14 q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 336 q - 2 q^{2} + 14 q^{3} - 26 q^{4} + 2 q^{5} + q^{6} - 4 q^{7} + 14 q^{9} + q^{10} - 2 q^{11} - 182 q^{12} - 78 q^{13} - 38 q^{14} - 24 q^{15} - 14 q^{16} - 3 q^{17} + q^{18} - 2 q^{19} - q^{20} + 2 q^{21} + 2 q^{22} - 7 q^{23} + 124 q^{25} + 118 q^{26} - 28 q^{27} - 80 q^{28} + 49 q^{29} + q^{30} - 12 q^{31} - 12 q^{32} + 11 q^{33} - 63 q^{34} + 65 q^{35} + 13 q^{36} - 25 q^{37} - 5 q^{38} + 73 q^{40} - 94 q^{41} - 7 q^{42} - 10 q^{43} - 12 q^{44} - 17 q^{45} - 16 q^{46} + 29 q^{47} + 7 q^{48} - 70 q^{49} - 36 q^{50} + 101 q^{51} - 27 q^{52} - 32 q^{53} - 2 q^{54} - 23 q^{55} - 130 q^{56} - 2 q^{57} + 302 q^{58} - 18 q^{59} - q^{60} - 40 q^{61} - 130 q^{62} + 28 q^{63} - 16 q^{64} + 52 q^{65} - 24 q^{66} - 114 q^{67} + 150 q^{68} - 3 q^{69} - 15 q^{70} + 27 q^{71} - 3 q^{73} - 125 q^{74} - 40 q^{75} + 368 q^{76} - 24 q^{77} - 41 q^{78} + 55 q^{79} + 78 q^{80} + 14 q^{81} + 24 q^{82} + 40 q^{83} + 105 q^{84} - 142 q^{85} - 48 q^{86} - 5 q^{87} - 255 q^{88} - 52 q^{89} - 2 q^{90} + 130 q^{91} - 366 q^{92} - 54 q^{93} - 64 q^{94} - 58 q^{95} + 136 q^{96} - 110 q^{97} - 6 q^{98} + 4 q^{99} + O(q^{100}) \)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

Label \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
19.1 −2.67728 0.659891i −0.845190 0.534466i 4.96148 + 2.60399i 2.70271 2.03095i 1.91013 + 1.98865i 0.750173 0.393721i −7.43705 6.58866i 0.428693 + 0.903450i −8.57612 + 3.65394i
19.2 −2.54821 0.628078i −0.845190 0.534466i 4.32800 + 2.27151i −2.61874 + 1.96786i 1.81804 + 1.89278i 1.20275 0.631251i −5.67309 5.02592i 0.428693 + 0.903450i 7.90908 3.36974i
19.3 −2.06747 0.509586i −0.845190 0.534466i 2.24385 + 1.17766i −0.929494 + 0.698469i 1.47505 + 1.53569i −0.0122606 + 0.00643484i −0.851290 0.754178i 0.428693 + 0.903450i 2.27763 0.970408i
19.4 −1.84576 0.454940i −0.845190 0.534466i 1.42896 + 0.749975i 1.98261 1.48983i 1.31687 + 1.37101i −1.31241 + 0.688808i 0.549517 + 0.486830i 0.428693 + 0.903450i −4.33720 + 1.84791i
19.5 −1.32814 0.327356i −0.845190 0.534466i −0.114131 0.0599004i −1.19669 + 0.899253i 0.947566 + 0.986521i −3.33061 + 1.74804i 2.17972 + 1.93107i 0.428693 + 0.903450i 1.88374 0.802587i
19.6 −0.670127 0.165171i −0.845190 0.534466i −1.34912 0.708075i 2.10094 1.57875i 0.478106 + 0.497761i 2.63145 1.38109i 1.82035 + 1.61269i 0.428693 + 0.903450i −1.66866 + 0.710950i
19.7 −0.252108 0.0621390i −0.845190 0.534466i −1.71121 0.898115i 0.591924 0.444802i 0.179868 + 0.187262i −1.74335 + 0.914980i 0.764309 + 0.677118i 0.428693 + 0.903450i −0.176868 + 0.0753565i
19.8 0.304362 + 0.0750185i −0.845190 0.534466i −1.68390 0.883781i −1.40241 + 1.05384i −0.217149 0.226076i 3.02560 1.58796i −0.915489 0.811052i 0.428693 + 0.903450i −0.505897 + 0.215543i
19.9 0.309649 + 0.0763217i −0.845190 0.534466i −1.68085 0.882180i −2.49078 + 1.87170i −0.220921 0.230003i 1.37070 0.719396i −0.930571 0.824414i 0.428693 + 0.903450i −0.914121 + 0.389471i
19.10 1.12662 + 0.277686i −0.845190 0.534466i −0.578755 0.303754i 2.59048 1.94662i −0.803792 0.836836i −2.02054 + 1.06046i −2.30473 2.04182i 0.428693 + 0.903450i 3.45903 1.47376i
19.11 1.92606 + 0.474730i −0.845190 0.534466i 1.71341 + 0.899266i −2.98759 + 2.24503i −1.37416 1.43065i −2.68643 + 1.40995i −0.0964276 0.0854274i 0.428693 + 0.903450i −6.82004 + 2.90575i
19.12 1.97256 + 0.486192i −0.845190 0.534466i 1.88370 + 0.988640i 2.48102 1.86436i −1.40733 1.46519i 1.36867 0.718334i 0.193693 + 0.171597i 0.428693 + 0.903450i 5.80039 2.47132i
19.13 2.10415 + 0.518627i −0.845190 0.534466i 2.38758 + 1.25310i −0.272381 + 0.204681i −1.50122 1.56294i 2.92878 1.53714i 1.12970 + 1.00083i 0.428693 + 0.903450i −0.679283 + 0.289416i
19.14 2.67476 + 0.659270i −0.845190 0.534466i 4.94881 + 2.59734i 0.358425 0.269339i −1.90833 1.98678i −1.72808 + 0.906968i 7.40054 + 6.55631i 0.428693 + 0.903450i 1.13627 0.484119i
37.1 −2.29227 + 1.20308i −0.996757 + 0.0804666i 2.67098 3.86958i −0.825574 + 0.859513i 2.18803 1.38363i −0.656511 0.951121i −0.843111 + 6.94364i 0.987050 0.160411i 0.858377 2.96346i
37.2 −1.69947 + 0.891951i −0.996757 + 0.0804666i 0.956496 1.38572i −2.86891 + 2.98685i 1.62219 1.02581i −1.57479 2.28148i 0.0731569 0.602501i 0.987050 0.160411i 2.21150 7.63499i
37.3 −1.68163 + 0.882587i −0.996757 + 0.0804666i 0.912786 1.32240i −1.29736 + 1.35069i 1.60516 1.01504i 1.91693 + 2.77716i 0.0900034 0.741244i 0.987050 0.160411i 0.989572 3.41640i
37.4 −1.44364 + 0.757679i −0.996757 + 0.0804666i 0.373879 0.541657i 1.60746 1.67354i 1.37799 0.871386i −2.41370 3.49684i 0.263700 2.17176i 0.987050 0.160411i −1.05258 + 3.63393i
37.5 −1.33614 + 0.701259i −0.996757 + 0.0804666i 0.157371 0.227991i 2.79334 2.90817i 1.27538 0.806500i 1.85954 + 2.69400i 0.313387 2.58098i 0.987050 0.160411i −1.69290 + 5.84458i
37.6 −0.369506 + 0.193932i −0.996757 + 0.0804666i −1.03720 + 1.50265i 0.841882 0.876493i 0.352702 0.223035i −0.419250 0.607388i 0.192443 1.58491i 0.987050 0.160411i −0.141101 + 0.487136i
See next 80 embeddings (of 336 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 427.14
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
157.i even 39 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 471.2.q.b 336
157.i even 39 1 inner 471.2.q.b 336
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
471.2.q.b 336 1.a even 1 1 trivial
471.2.q.b 336 157.i even 39 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(38\!\cdots\!81\)\( T_{2}^{314} + \)\(79\!\cdots\!22\)\( T_{2}^{313} + \)\(34\!\cdots\!11\)\( T_{2}^{312} + \)\(70\!\cdots\!87\)\( T_{2}^{311} + \)\(30\!\cdots\!78\)\( T_{2}^{310} + \)\(60\!\cdots\!57\)\( T_{2}^{309} + \)\(26\!\cdots\!83\)\( T_{2}^{308} + \)\(51\!\cdots\!50\)\( T_{2}^{307} + \)\(21\!\cdots\!68\)\( T_{2}^{306} + \)\(42\!\cdots\!39\)\( T_{2}^{305} + \)\(17\!\cdots\!93\)\( T_{2}^{304} + \)\(34\!\cdots\!69\)\( T_{2}^{303} + \)\(14\!\cdots\!68\)\( T_{2}^{302} + \)\(27\!\cdots\!74\)\( T_{2}^{301} + \)\(11\!\cdots\!75\)\( T_{2}^{300} + \)\(21\!\cdots\!42\)\( T_{2}^{299} + \)\(86\!\cdots\!71\)\( T_{2}^{298} + \)\(16\!\cdots\!95\)\( T_{2}^{297} + \)\(64\!\cdots\!66\)\( T_{2}^{296} + \)\(11\!\cdots\!30\)\( T_{2}^{295} + \)\(47\!\cdots\!58\)\( T_{2}^{294} + \)\(87\!\cdots\!08\)\( T_{2}^{293} + \)\(34\!\cdots\!09\)\( T_{2}^{292} + \)\(62\!\cdots\!73\)\( T_{2}^{291} + \)\(24\!\cdots\!76\)\( T_{2}^{290} + \)\(43\!\cdots\!71\)\( T_{2}^{289} + \)\(17\!\cdots\!94\)\( T_{2}^{288} + \)\(30\!\cdots\!71\)\( T_{2}^{287} + \)\(12\!\cdots\!32\)\( T_{2}^{286} + \)\(20\!\cdots\!30\)\( T_{2}^{285} + \)\(82\!\cdots\!67\)\( T_{2}^{284} + \)\(13\!\cdots\!98\)\( T_{2}^{283} + \)\(55\!\cdots\!10\)\( T_{2}^{282} + \)\(90\!\cdots\!14\)\( T_{2}^{281} + \)\(36\!\cdots\!89\)\( T_{2}^{280} + \)\(58\!\cdots\!92\)\( T_{2}^{279} + \)\(23\!\cdots\!61\)\( T_{2}^{278} + \)\(36\!\cdots\!51\)\( T_{2}^{277} + \)\(15\!\cdots\!24\)\( T_{2}^{276} + \)\(23\!\cdots\!40\)\( T_{2}^{275} + \)\(94\!\cdots\!57\)\( T_{2}^{274} + \)\(14\!\cdots\!24\)\( T_{2}^{273} + \)\(58\!\cdots\!22\)\( T_{2}^{272} + \)\(84\!\cdots\!04\)\( T_{2}^{271} + \)\(35\!\cdots\!53\)\( T_{2}^{270} + \)\(50\!\cdots\!10\)\( T_{2}^{269} + \)\(21\!\cdots\!57\)\( T_{2}^{268} + \)\(29\!\cdots\!30\)\( T_{2}^{267} + \)\(12\!\cdots\!89\)\( T_{2}^{266} + \)\(16\!\cdots\!11\)\( T_{2}^{265} + \)\(73\!\cdots\!49\)\( T_{2}^{264} + \)\(95\!\cdots\!89\)\( T_{2}^{263} + \)\(42\!\cdots\!84\)\( T_{2}^{262} + \)\(52\!\cdots\!44\)\( T_{2}^{261} + \)\(23\!\cdots\!51\)\( T_{2}^{260} + \)\(28\!\cdots\!12\)\( T_{2}^{259} + \)\(13\!\cdots\!67\)\( T_{2}^{258} + \)\(15\!\cdots\!96\)\( T_{2}^{257} + \)\(72\!\cdots\!53\)\( T_{2}^{256} + \)\(81\!\cdots\!58\)\( T_{2}^{255} + \)\(38\!\cdots\!90\)\( T_{2}^{254} + \)\(42\!\cdots\!26\)\( T_{2}^{253} + \)\(20\!\cdots\!91\)\( T_{2}^{252} + \)\(21\!\cdots\!54\)\( T_{2}^{251} + \)\(10\!\cdots\!63\)\( T_{2}^{250} + \)\(10\!\cdots\!41\)\( T_{2}^{249} + \)\(54\!\cdots\!97\)\( T_{2}^{248} + \)\(53\!\cdots\!73\)\( T_{2}^{247} + \)\(27\!\cdots\!06\)\( T_{2}^{246} + \)\(26\!\cdots\!52\)\( T_{2}^{245} + \)\(13\!\cdots\!81\)\( T_{2}^{244} + \)\(12\!\cdots\!36\)\( T_{2}^{243} + \)\(67\!\cdots\!67\)\( T_{2}^{242} + \)\(59\!\cdots\!29\)\( T_{2}^{241} + \)\(32\!\cdots\!50\)\( T_{2}^{240} + \)\(28\!\cdots\!03\)\( T_{2}^{239} + \)\(15\!\cdots\!40\)\( T_{2}^{238} + \)\(13\!\cdots\!97\)\( T_{2}^{237} + \)\(71\!\cdots\!00\)\( T_{2}^{236} + \)\(60\!\cdots\!91\)\( T_{2}^{235} + \)\(32\!\cdots\!84\)\( T_{2}^{234} + \)\(27\!\cdots\!91\)\( T_{2}^{233} + \)\(14\!\cdots\!03\)\( T_{2}^{232} + \)\(12\!\cdots\!45\)\( T_{2}^{231} + \)\(66\!\cdots\!64\)\( T_{2}^{230} + \)\(53\!\cdots\!43\)\( T_{2}^{229} + \)\(29\!\cdots\!14\)\( T_{2}^{228} + \)\(23\!\cdots\!49\)\( T_{2}^{227} + \)\(12\!\cdots\!09\)\( T_{2}^{226} + \)\(10\!\cdots\!96\)\( T_{2}^{225} + \)\(54\!\cdots\!67\)\( T_{2}^{224} + \)\(43\!\cdots\!49\)\( T_{2}^{223} + \)\(22\!\cdots\!76\)\( T_{2}^{222} + \)\(18\!\cdots\!68\)\( T_{2}^{221} + \)\(94\!\cdots\!32\)\( T_{2}^{220} + \)\(77\!\cdots\!06\)\( T_{2}^{219} + \)\(38\!\cdots\!38\)\( T_{2}^{218} + \)\(32\!\cdots\!91\)\( T_{2}^{217} + \)\(15\!\cdots\!74\)\( T_{2}^{216} + \)\(13\!\cdots\!31\)\( T_{2}^{215} + \)\(61\!\cdots\!90\)\( T_{2}^{214} + \)\(53\!\cdots\!35\)\( T_{2}^{213} + \)\(24\!\cdots\!77\)\( T_{2}^{212} + \)\(21\!\cdots\!18\)\( T_{2}^{211} + \)\(93\!\cdots\!90\)\( T_{2}^{210} + \)\(83\!\cdots\!73\)\( T_{2}^{209} + \)\(35\!\cdots\!78\)\( T_{2}^{208} + \)\(32\!\cdots\!93\)\( T_{2}^{207} + \)\(13\!\cdots\!34\)\( T_{2}^{206} + \)\(12\!\cdots\!85\)\( T_{2}^{205} + \)\(49\!\cdots\!04\)\( T_{2}^{204} + \)\(46\!\cdots\!94\)\( T_{2}^{203} + \)\(17\!\cdots\!77\)\( T_{2}^{202} + \)\(17\!\cdots\!33\)\( T_{2}^{201} + \)\(64\!\cdots\!83\)\( T_{2}^{200} + \)\(63\!\cdots\!29\)\( T_{2}^{199} + \)\(22\!\cdots\!70\)\( T_{2}^{198} + \)\(23\!\cdots\!47\)\( T_{2}^{197} + \)\(78\!\cdots\!50\)\( T_{2}^{196} + \)\(81\!\cdots\!49\)\( T_{2}^{195} + \)\(26\!\cdots\!44\)\( T_{2}^{194} + \)\(28\!\cdots\!77\)\( T_{2}^{193} + \)\(90\!\cdots\!40\)\( T_{2}^{192} + \)\(97\!\cdots\!73\)\( T_{2}^{191} + \)\(29\!\cdots\!39\)\( T_{2}^{190} + \)\(32\!\cdots\!73\)\( T_{2}^{189} + \)\(96\!\cdots\!91\)\( T_{2}^{188} + \)\(10\!\cdots\!53\)\( T_{2}^{187} + \)\(30\!\cdots\!29\)\( T_{2}^{186} + \)\(35\!\cdots\!60\)\( T_{2}^{185} + \)\(96\!\cdots\!10\)\( T_{2}^{184} + \)\(11\!\cdots\!47\)\( T_{2}^{183} + \)\(29\!\cdots\!47\)\( T_{2}^{182} + \)\(34\!\cdots\!88\)\( T_{2}^{181} + \)\(88\!\cdots\!23\)\( T_{2}^{180} + \)\(10\!\cdots\!50\)\( T_{2}^{179} + \)\(26\!\cdots\!94\)\( T_{2}^{178} + \)\(31\!\cdots\!78\)\( T_{2}^{177} + \)\(74\!\cdots\!28\)\( T_{2}^{176} + \)\(90\!\cdots\!20\)\( T_{2}^{175} + \)\(20\!\cdots\!06\)\( T_{2}^{174} + \)\(25\!\cdots\!16\)\( T_{2}^{173} + \)\(55\!\cdots\!42\)\( T_{2}^{172} + \)\(68\!\cdots\!27\)\( T_{2}^{171} + \)\(14\!\cdots\!86\)\( T_{2}^{170} + \)\(18\!\cdots\!98\)\( T_{2}^{169} + \)\(37\!\cdots\!20\)\( T_{2}^{168} + \)\(46\!\cdots\!51\)\( T_{2}^{167} + \)\(91\!\cdots\!76\)\( T_{2}^{166} + \)\(11\!\cdots\!45\)\( T_{2}^{165} + \)\(21\!\cdots\!14\)\( T_{2}^{164} + \)\(27\!\cdots\!53\)\( T_{2}^{163} + \)\(50\!\cdots\!79\)\( T_{2}^{162} + \)\(62\!\cdots\!55\)\( T_{2}^{161} + \)\(11\!\cdots\!87\)\( T_{2}^{160} + \)\(14\!\cdots\!08\)\( T_{2}^{159} + \)\(25\!\cdots\!71\)\( T_{2}^{158} + \)\(30\!\cdots\!06\)\( T_{2}^{157} + \)\(53\!\cdots\!13\)\( T_{2}^{156} + \)\(66\!\cdots\!89\)\( T_{2}^{155} + \)\(11\!\cdots\!99\)\( T_{2}^{154} + \)\(13\!\cdots\!57\)\( T_{2}^{153} + \)\(23\!\cdots\!53\)\( T_{2}^{152} + \)\(28\!\cdots\!12\)\( T_{2}^{151} + \)\(46\!\cdots\!58\)\( T_{2}^{150} + \)\(56\!\cdots\!96\)\( T_{2}^{149} + \)\(90\!\cdots\!55\)\( T_{2}^{148} + \)\(10\!\cdots\!00\)\( T_{2}^{147} + \)\(17\!\cdots\!89\)\( T_{2}^{146} + \)\(20\!\cdots\!28\)\( T_{2}^{145} + \)\(32\!\cdots\!79\)\( T_{2}^{144} + \)\(37\!\cdots\!30\)\( T_{2}^{143} + \)\(57\!\cdots\!02\)\( T_{2}^{142} + \)\(66\!\cdots\!69\)\( T_{2}^{141} + \)\(98\!\cdots\!27\)\( T_{2}^{140} + \)\(11\!\cdots\!81\)\( T_{2}^{139} + \)\(16\!\cdots\!63\)\( T_{2}^{138} + \)\(18\!\cdots\!80\)\( T_{2}^{137} + \)\(26\!\cdots\!13\)\( T_{2}^{136} + \)\(28\!\cdots\!73\)\( T_{2}^{135} + \)\(40\!\cdots\!62\)\( T_{2}^{134} + \)\(43\!\cdots\!71\)\( T_{2}^{133} + \)\(61\!\cdots\!68\)\( T_{2}^{132} + \)\(64\!\cdots\!68\)\( T_{2}^{131} + \)\(91\!\cdots\!19\)\( T_{2}^{130} + \)\(92\!\cdots\!80\)\( T_{2}^{129} + \)\(13\!\cdots\!27\)\( T_{2}^{128} + \)\(13\!\cdots\!69\)\( T_{2}^{127} + \)\(19\!\cdots\!19\)\( T_{2}^{126} + \)\(18\!\cdots\!31\)\( T_{2}^{125} + \)\(26\!\cdots\!58\)\( T_{2}^{124} + \)\(25\!\cdots\!46\)\( T_{2}^{123} + \)\(36\!\cdots\!68\)\( T_{2}^{122} + \)\(33\!\cdots\!00\)\( T_{2}^{121} + \)\(48\!\cdots\!88\)\( T_{2}^{120} + \)\(43\!\cdots\!99\)\( T_{2}^{119} + \)\(63\!\cdots\!71\)\( T_{2}^{118} + \)\(55\!\cdots\!86\)\( T_{2}^{117} + \)\(80\!\cdots\!21\)\( T_{2}^{116} + \)\(68\!\cdots\!34\)\( T_{2}^{115} + \)\(99\!\cdots\!72\)\( T_{2}^{114} + \)\(82\!\cdots\!38\)\( T_{2}^{113} + \)\(12\!\cdots\!95\)\( T_{2}^{112} + \)\(97\!\cdots\!71\)\( T_{2}^{111} + \)\(14\!\cdots\!79\)\( T_{2}^{110} + \)\(11\!\cdots\!97\)\( T_{2}^{109} + \)\(16\!\cdots\!31\)\( T_{2}^{108} + \)\(12\!\cdots\!44\)\( T_{2}^{107} + \)\(19\!\cdots\!55\)\( T_{2}^{106} + \)\(14\!\cdots\!21\)\( T_{2}^{105} + \)\(20\!\cdots\!50\)\( T_{2}^{104} + \)\(16\!\cdots\!50\)\( T_{2}^{103} + \)\(22\!\cdots\!83\)\( T_{2}^{102} + \)\(17\!\cdots\!83\)\( T_{2}^{101} + \)\(22\!\cdots\!71\)\( T_{2}^{100} + \)\(18\!\cdots\!22\)\( T_{2}^{99} + \)\(22\!\cdots\!21\)\( T_{2}^{98} + \)\(18\!\cdots\!92\)\( T_{2}^{97} + \)\(21\!\cdots\!09\)\( T_{2}^{96} + \)\(18\!\cdots\!38\)\( T_{2}^{95} + \)\(19\!\cdots\!35\)\( T_{2}^{94} + \)\(17\!\cdots\!93\)\( T_{2}^{93} + \)\(17\!\cdots\!90\)\( T_{2}^{92} + \)\(15\!\cdots\!05\)\( T_{2}^{91} + \)\(14\!\cdots\!45\)\( T_{2}^{90} + \)\(12\!\cdots\!68\)\( T_{2}^{89} + \)\(11\!\cdots\!68\)\( T_{2}^{88} + \)\(99\!\cdots\!40\)\( T_{2}^{87} + \)\(86\!\cdots\!10\)\( T_{2}^{86} + \)\(72\!\cdots\!47\)\( T_{2}^{85} + \)\(61\!\cdots\!06\)\( T_{2}^{84} + \)\(50\!\cdots\!70\)\( T_{2}^{83} + \)\(40\!\cdots\!98\)\( T_{2}^{82} + \)\(32\!\cdots\!41\)\( T_{2}^{81} + \)\(25\!\cdots\!28\)\( T_{2}^{80} + \)\(19\!\cdots\!12\)\( T_{2}^{79} + \)\(14\!\cdots\!81\)\( T_{2}^{78} + \)\(10\!\cdots\!31\)\( T_{2}^{77} + \)\(75\!\cdots\!55\)\( T_{2}^{76} + \)\(53\!\cdots\!54\)\( T_{2}^{75} + \)\(36\!\cdots\!13\)\( T_{2}^{74} + \)\(24\!\cdots\!32\)\( T_{2}^{73} + \)\(15\!\cdots\!43\)\( T_{2}^{72} + \)\(10\!\cdots\!54\)\( T_{2}^{71} + \)\(61\!\cdots\!26\)\( T_{2}^{70} + \)\(35\!\cdots\!02\)\( T_{2}^{69} + \)\(20\!\cdots\!54\)\( T_{2}^{68} + \)\(10\!\cdots\!07\)\( T_{2}^{67} + \)\(52\!\cdots\!58\)\( T_{2}^{66} + \)\(23\!\cdots\!55\)\( T_{2}^{65} + \)\(10\!\cdots\!62\)\( T_{2}^{64} + \)\(35\!\cdots\!85\)\( T_{2}^{63} + \)\(11\!\cdots\!17\)\( T_{2}^{62} + \)\(25\!\cdots\!34\)\( T_{2}^{61} + \)\(37\!\cdots\!46\)\( T_{2}^{60} - \)\(21\!\cdots\!74\)\( T_{2}^{59} + \)\(44\!\cdots\!38\)\( T_{2}^{58} + \)\(73\!\cdots\!95\)\( T_{2}^{57} + \)\(32\!\cdots\!24\)\( T_{2}^{56} - \)\(79\!\cdots\!70\)\( T_{2}^{55} + \)\(38\!\cdots\!47\)\( T_{2}^{54} - \)\(41\!\cdots\!28\)\( T_{2}^{53} + \)\(15\!\cdots\!28\)\( T_{2}^{52} - \)\(30\!\cdots\!42\)\( T_{2}^{51} + \)\(56\!\cdots\!34\)\( T_{2}^{50} - \)\(12\!\cdots\!59\)\( T_{2}^{49} + \)\(42\!\cdots\!89\)\( T_{2}^{48} - \)\(47\!\cdots\!79\)\( T_{2}^{47} + \)\(10\!\cdots\!65\)\( T_{2}^{46} - \)\(30\!\cdots\!34\)\( T_{2}^{45} + \)\(30\!\cdots\!25\)\( T_{2}^{44} - \)\(69\!\cdots\!60\)\( T_{2}^{43} + \)\(16\!\cdots\!75\)\( T_{2}^{42} - \)\(14\!\cdots\!05\)\( T_{2}^{41} + \)\(40\!\cdots\!46\)\( T_{2}^{40} - \)\(53\!\cdots\!73\)\( T_{2}^{39} + \)\(38\!\cdots\!27\)\( T_{2}^{38} - \)\(17\!\cdots\!28\)\( T_{2}^{37} + \)\(18\!\cdots\!83\)\( T_{2}^{36} - \)\(30\!\cdots\!74\)\( T_{2}^{35} + \)\(46\!\cdots\!68\)\( T_{2}^{34} - \)\(10\!\cdots\!74\)\( T_{2}^{33} - \)\(38\!\cdots\!58\)\( T_{2}^{32} - \)\(49\!\cdots\!46\)\( T_{2}^{31} + \)\(34\!\cdots\!82\)\( T_{2}^{30} - \)\(26\!\cdots\!65\)\( T_{2}^{29} - \)\(57\!\cdots\!34\)\( T_{2}^{28} - \)\(23\!\cdots\!26\)\( T_{2}^{27} + \)\(11\!\cdots\!56\)\( T_{2}^{26} - \)\(92\!\cdots\!62\)\( T_{2}^{25} + \)\(52\!\cdots\!17\)\( T_{2}^{24} - \)\(85\!\cdots\!80\)\( T_{2}^{23} + \)\(28\!\cdots\!76\)\( T_{2}^{22} - \)\(52\!\cdots\!28\)\( T_{2}^{21} + \)\(15\!\cdots\!28\)\( T_{2}^{20} - \)\(53\!\cdots\!92\)\( T_{2}^{19} + \)\(14\!\cdots\!56\)\( T_{2}^{18} - \)\(29\!\cdots\!16\)\( T_{2}^{17} + \)\(54\!\cdots\!56\)\( T_{2}^{16} - \)\(11\!\cdots\!16\)\( T_{2}^{15} + \)\(24\!\cdots\!64\)\( T_{2}^{14} - \)\(46\!\cdots\!68\)\( T_{2}^{13} + \)\(74\!\cdots\!48\)\( T_{2}^{12} - \)\(94\!\cdots\!36\)\( T_{2}^{11} + \)\(95\!\cdots\!16\)\( T_{2}^{10} - \)\(74\!\cdots\!20\)\( T_{2}^{9} + \)\(45\!\cdots\!44\)\( T_{2}^{8} - \)\(20\!\cdots\!12\)\( T_{2}^{7} + \)\(62\!\cdots\!92\)\( T_{2}^{6} - \)\(12\!\cdots\!72\)\( T_{2}^{5} + \)\(16\!\cdots\!92\)\( T_{2}^{4} - \)\(12\!\cdots\!16\)\( T_{2}^{3} + \)\(53\!\cdots\!56\)\( T_{2}^{2} - \)\(36\!\cdots\!56\)\( T_{2} + \)\(33\!\cdots\!84\)\( \)">\(T_{2}^{336} + \cdots\) acting on \(S_{2}^{\mathrm{new}}(471, [\chi])\).